Compact Composition operators
on L^1
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Joel H. Shapiro and Carl Sundberg
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Proc. Amer. Math.
Soc. 108 (1990), 443--449.
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| Abstract: The composition operator induced by a holomorphic
selfmap of the unit disc is compact on L^1 of the unit circle
if and only if it is compact on the Hardy space H^2 of the disc.
This shows that an integral condition of Sarason characterizing
compactness on L^1 is equivalent to Shapiro's asymptotic condition
on the Nevanlinna counting function that characterizes compactness
on H^2 (The essential norm of a composition operator,
Annals of Mathematics125 (1987), 375--404). |
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